Source of proof of $\mathbb{E}[X] = \int_{0}^{\infty}(1 - F(x))dx - \int_{-\infty}^{0} F(x) dx$ Wikipedia provides a proof in their 'Basic Properties' section. I would prefer not to cite Wikipedia for this proof if there exists a published work giving the same result. Is there a proper source that I can cite?
 A: That is a consequence of the layer cake identity for measure spaces (https://en.wikipedia.org/wiki/Layer_cake_representation). In probability notation, it says that if $0 \leq X \leq \infty$ is a random variable, then
$$E(X) = \int_{0}^{\infty}(1 - F(x))\,dx.$$
It is a consequence because if $X \in L^1$, then write $X = X^+ - X^-$, $E(X) = E(X^+) - E(X^-)$, and apply the layer cake identity to $X^+$ and $X^-$ to get the result.
The layer cake identity itself is an easy consequence of Tonelli's theorem:
\begin{align}
E(X) &= \int_{\Omega}X(\omega)\,dP(\omega) \\
&= \int_{\Omega}\int_{0}^{X(\omega)}1\,dx\,dP(\omega) \\
&= \int_{\Omega}\int_{0}^{\infty}I(x < X(\omega))\,dx\,dP(\omega) \\
&= \int_{0}^{\infty}\int_{\Omega}I(x < X(\omega))\,dP(\omega)\,dx \\
&= \int_{0}^{\infty}P(X > x)\,dx.
\end{align}
A: You can get the result more simply by appealing to integration by parts of the Lebesgue-Stieltes integral $\ \int x\,dF(x)\ $. Volume $2$ of Feller's An Introduction to Probability Theory and applications has a proof of one half of the more general result
$$
\int_{-\infty}^\infty x^\alpha dF(x)=\alpha\int_0^\infty x^{\alpha-1}(1-F(x))dx-\alpha\int_{-\infty}^0x^{\alpha-1}F(x)dx\ .
$$
His Lemma $1$ on p.$148$ is a statement of the right tail half of this result, namely,
$$
\int_0^\infty x^\alpha dF(x)=\alpha\int_0^\infty x^{\alpha-1}(1-F(x))dx\ ,
$$
for which he gives a proof.  He doesn't state or prove the left tail half,
$$
\int_{-\infty}^a x^\alpha dF(x)=-\alpha\int_{-\infty}^0x^{\alpha-1}F(x)dx\ ,
$$
but merely notes that an analogous result holds for the left tail.
